Type classes

Philip Wadler

Implicit programming (IP) mechanisms infer values by type-directed resolution, making programs more compact and easier to read. Examples of IP mechanisms include Haskell’s type classes, Scala’s implicits, Agda’s instance arguments, Coq’s type classes and Rust’s traits. The design of IP mechanisms has led to heated debate: proponents of one school argue for the desirability of strong reasoning properties, while proponents of another school argue for the power and flexibility of local scoping or overlapping instances. The current state of affairs seems to indicate that the two goals are at odds with one another and cannot easily be reconciled. This paper presents COCHIS, the Calculus Of CoHerent ImplicitS, an improved variant of the implicit calculus that offers flexibility while preserving two key properties: coherence and stability of type substitutions. COCHIS supports polymorphism, local scoping, overlapping instances, first-class instances and higher-order rules, while remaining type-safe, coherent and stable under type substitution. We introduce a logical formulation of how to resolve implicits, which is simple but ambiguous and incoherent, and a second formulation, which is less simple but unambiguous, coherent and stable. Every resolution of the second formulation is also a resolution of the first, but not conversely. Parts of the second formulation bear a close resemblance to a standard technique for proof search called focusing. Moreover, key for its coherence is a rigorous enforcement of determinism.

Quantified class constraints have been proposed many years ago to raise the expressive power of type classes from Horn clauses to the universal fragment of Hereditiary Harrop logic. Yet, while it has been much asked for over the years, the feature was never implemented or studied in depth. Instead, several workarounds have been proposed, all of which are ultimately stopgap measures.

This paper revisits the idea of quantified class constraints and elaborates it into a practical language design. We show the merit of quantified class constraints in terms of more expressive modeling and in terms of terminating type class resolution. In addition, we provide a declarative specification of the type system as well as a type inference algorithm that elaborates into System F. Moreover, we discuss termination conditions of our system and also provide a prototype implementation.

Generic programming (GP) is an increasingly important trend in programming languages. Well-known GP mechanisms, such as type classes and the C++0x concepts proposal, usually combine two features: 1) a special type of interfaces; and 2) implicit instantiation of implementations of those interfaces.

Scala implicits are a GP language mechanism, inspired by type classes, that break with the tradition of coupling implicit instantiation with a special type of interface. Instead, implicits provide only implicit instantiation, which is generalized to work for any types. Scala implicits turn out to be quite powerful and useful to address many limitations that show up in other GP mechanisms.

This paper synthesizes the key ideas of implicits formally in a minimal and general core calculus called the implicit calculus (\lambda_?), and it shows how to build source languages supporting implicit instantiation on top of it. A novelty of the calculus is its support for partial resolution and higher-order rules (a feature that has been proposed before, but was never formalized or implemented). Ultimately, the implicit calculus provides a formal model of implicits, which can be used by language designers to study and inform implementations of similar mechanisms in their own languages.

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We study a minimal extension of the Hindley/Milner system that supports overloading and polymorphic records. We show that the type system is sound with respect to a standard untyped compositional semantics. We also show that every typable term in this system has a principal type and give an algorithm to reconstruct that type.

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This paper defines a set of type inference rules for resolving overloading introduced by type classes. Programs including type classes are transformed into ones which may be typed by the Hindley-Milner inference rules. In contrast to an other work on type classes, the rules presented here relate directly to user programs. An innovative aspect of this work is the use of second-order lambda calculus to record type information in the program.

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This paper presents type classes, a new approach to ad-hoc polymorphism. Type classes permit overloading of arithmetic operators such as multiplication, and generalise the ``eqtype variables'' of Standard ML. Type classes extend the Hindley-Milner polymorphic type system, and provide a new approach to issues that arise in object-oriented programming, bounded type quantification, and abstract data types. This paper provides an informal introduction to type classes, and defines them formally by means of type inference rules.

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Proposal: Overloading in Haskell

[The original proposal for type classes. Never published.]

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Philip Wadler,